Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×#11764049)⋊C4 |
| Fitting | C2×C4×C137 ⋊ C8 |
| Sylow | C137 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11764049 |
| commutator | C2×C137 | C4×C8 |
| Fitting | C2×C4×C137 | C8 |
| socle | C22×C137 | C2×C8 |
| hypercenter | C2×C4 | #11764049 |
| hypocenter | C137 | #336 |
| LCS | G ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
118 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=d,b^2=e,d^2=f,b^a=bc,g^a=g^127,g^d=g^100,g^f=g^136176 conjugacy classes — view the full character table.
| maximal subgroup of | #31797788, #31797789, #31797790, … |
| maximal normal subgroup of | #31797788, #31797789, #31797790, … |
| maximal quotient of | (C2×#11764049)⋊C8, #31797792, #31797793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1640);